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 A048779 Coefficients of power series for (1-(1-8x)^(1/4))/2. 5
 1, 3, 14, 77, 462, 2926, 19228, 129789, 894102, 6258714, 44379972, 318056466, 2299792908, 16755634044, 122874649656, 906200541213, 6716545187814, 50000947509282, 373691291911476, 2802684689336070, 21086865757861860, 159109987082048580, 1203701641403324040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..1112 ElÅ¼bieta Liszewska, Wojciech MÅ‚otkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. FORMULA a(n) = 2^(n-1)*3*7*11*...*(4n-5)/n! = 2a(n-1)(32a(n-2)+a(n-1))/(18a(n-2)-a(n-1)). a(n) = -A004984(n)/2. n*a(n) + 2*(5-4*n)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012 G.f. A(x) =: y satisfies x = y * (1 - y) * (1 - 2*y + 2*y^2). - Michael Somos, Jan 17 2014 0 = a(n) * (64*a(n+1) - 18*a(n+2)) + a(n+1) * (2*a(n+1) + a(n+2)) unless n=0. - Michael Somos, Jan 17 2014 a(n) = -A004984(n)/2 unless n=0. - Michael Somos, Jan 17 2014 From Karol A. Penson, Dec 19 2015: (Start) a(n) = 8^n*binomial(n-1/4,-1/4)/(n+1). E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [2], 8*x). Representation as n-th moment of a positive function on (0, 8): a(n)=int(x^n*((2^(1/4)/(2*Pi*x^(1/4))*(1-x/8)^(1/4))), x=0..8), n=0,1,... . This function is the solution of the Hausdorff moment problem on (0, 8) with moments equal to a(n). As a consequence this representation is unique. (End) EXAMPLE G.f.: x + 3*x^2 + 14*x^3 + 77*x^4 + 462*x^5 + 2926*x^6 + 19228*x^7 + ... MATHEMATICA a[ n_] := If[n < 1, 0, (-1/2) Pochhammer[ -1/4, n] 8^n/n!] (* Michael Somos, Jan 17 2014 *) a[ n_] := SeriesCoefficient[ (1 - (1 - 8 x)^(1/4)) / 2, {x, 0, n}] (* Michael Somos, Jan 17 2014 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - (1 - 8*x + x * O(x^n))^(1/4)) / 2, n))} /* Michael Somos, Jan 17 2014 */ CROSSREFS Related to Catalan numbers (A000108) which are coefficients for (1-(1-4x)^(1/2))/2. Cf. A004984. Sequence in context: A198649 A198656 A240402 * A052186 A330518 A228656 Adjacent sequences:  A048776 A048777 A048778 * A048780 A048781 A048782 KEYWORD nonn AUTHOR STATUS approved

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Last modified June 19 16:32 EDT 2021. Contains 345144 sequences. (Running on oeis4.)